Unclassified Report: Jacobi-Davidson methods and preconditioning with applications in pole-zero analysis

This report discusses the application of Jacobi-Davidson style methods in electric circuit simulation. Using the generalised eigenvalue problem, which arises from pole-zero analysis, as a starting point, both the JDQR-method and the JDQZ-method are studied. Although the JDQR-method (for the ordinary eigenproblem) and the JDQZ-method (for the generalised eigenproblem) are designed to converge fast to a few selected eigenvalues, they will be used to compute all eigenvalues. With the help of suitable preconditioners for the GMRES process, to solve the correction equation of the Jacobi-Davidson method, the eigenmethods are made more suitable for pole-zero analysis. Numerical experiments show that the Jacobi-Davidson methods can be used for pole-zero analysis. However, in a comparison with the direct Q R and QZ methods, the shortages in accuracy for certain implementations of iterative methods become visible. Here preconditioning techniques improve the performance of the Jacobi-Davidson methods. The Arnoldi method is considered as the iterative competitor of the JacobiDavidson methods. Besides applications in pole-zero analysis, the Jacobi-Davidson methods are of great use in stability analysis and periodic steady state analysis. An implementation of the iterative Jacobi-Davidson methods in Pstar, respecting the hierarchy, is possible, because no dense, full-dimensional matrix multiplications are involved. A description of the hierarchical algorithm in Pstar is given. This project has been a cooperation between Philips Electronic Design & Tools/Analogue Simulation (ED&T/AS) and the Utrecht University (UU). It has been executed under supervision of Dr. E.J.W. ter Maten (ED&T/AS) and Prof.Dr. H.A. van der Vorst (UU). c ©Koninklijke Philips Electronics N.V. 2002 iii 2002/817 Unclassified Report Conclusions: The most important conclusion is that Jacobi-Davidson style methods are suitable for application in pole-zero analysis under the following assumptions, apart from the dimension of the problem: the eigenspectrum of the generalised eigenproblem must be not too wide or an efficient preconditioner must be available. If one or both of these assumptions are not met, there is no special preference for Jacobi-Davidson style methods above the (restarted) Arnoldi method. On the contrary, with the typical convergence behaviour of Jacobi-Davidson in mind, the Arnoldi method should be chosen in that case. Nevertheless, if both assumptions are met, one can profit from the quadratic convergence of the Jacobi-Davidson style methods, combined with acceptable accuracy. Arguments that Jacobi-Davidson style methods are more robust than the Arnoldi method are in this case not valid, as numerical experiments have shown. Some of these arguments are based on the current Arnoldi implementation in Pstar; some improvements for the Arnoldi implementation in Pstar are proposed. Nevertheless, Jacobi-Davidson style methods are very applicable in stability analysis and periodic steady-state analysis, where only one or a few eigenvalues are needed. For this type of applications, Jacobi-Davidson style methods are preferred over the Arnoldi method. Furthermore, JacobiDavidson style methods are suitable for high dimensional problems because the spectrum can be searched part by part. The Arnoldi method does lack this property. Finally, the direct Q R and QZ method are superior in accuracy, robustness and efficiency for problems with relatively small dimensions. Even for larger problems, their performance is acceptable. The only disadvantage is that the direct methods do not fit in the hierarchical implementation of Pstar, while the iterative methods do. iv c ©Koninklijke Philips Electronics N.V. 2002